Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  impcon4bid Structured version   Visualization version   GIF version

Theorem impcon4bid 216
 Description: A variation on impbid 201 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.)
Hypotheses
Ref Expression
impcon4bid.1 (𝜑 → (𝜓𝜒))
impcon4bid.2 (𝜑 → (¬ 𝜓 → ¬ 𝜒))
Assertion
Ref Expression
impcon4bid (𝜑 → (𝜓𝜒))

Proof of Theorem impcon4bid
StepHypRef Expression
1 impcon4bid.1 . 2 (𝜑 → (𝜓𝜒))
2 impcon4bid.2 . . 3 (𝜑 → (¬ 𝜓 → ¬ 𝜒))
32con4d 113 . 2 (𝜑 → (𝜒𝜓))
41, 3impbid 201 1 (𝜑 → (𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 196 This theorem is referenced by:  con4bid  306  soisoi  6478  isomin  6487  alephdom  8787  nn0n0n1ge2b  11236  om2uzlt2i  12612  sadcaddlem  15017  isprm5  15257  pcdvdsb  15411  cvgdvgrat  37534
 Copyright terms: Public domain W3C validator